Intl. Trans. in Op. Res. xx (2019) 1–27
The multiperiod two-dimensional non-guillotine cutting stock problem with usable leftovers
E. G. Birgina,∗, O. C. Rom˜aoaand D. P. Ronconib
aDepartment of Computer Science, Institute of Mathematics and Statistics, University of S˜ao Paulo, Rua do Mat˜ao, 1010, Cidade Universit´aria, 05508-090, S˜ao Paulo, SP, Brazil
bDepartment of Production Engineering, EP-USP, University of S˜ao Paulo, Av. Prof. Almeida Prado, 128, Cidade Universit´aria, 05508-900, S˜ao Paulo SP, Brazil
E-mail: [email protected] [E. G. Birgin]; [email protected] [O. C. Rom˜ao];
[email protected] [D. P. Ronconi]
Received DD MMMM YYYY; received in revised form DD MMMM YYYY; accepted DD MMMM YYYY
Abstract
A mixed integer linear programming model for the two-dimensional non-guillotine cutting problem with usable leftovers was recently introduced in Andradeet al. [Journal of the Operational Research Society65, pp. 1649- 1663, 2014]. The problem consists in cutting a set of ordered items using a set of objects of minimum cost and, within the set of solutions of minimum cost, maximizing the value of the usable leftovers. Since the concept of usable leftovers assumes they can potentially be used to attend new arriving orders, the problem is extended to the multiperiod framework in this work. In this way, the decision at each instant does not minimize in a myopic way the cost of the objects required to attend the orders of the current instant; but it aims to minimize the overall cost of the objects up to the considered time horizon. Some variants of the proposed model are analyzed and numerical results are presented.
Keywords:Non-guillotine cutting and packing; usable leftovers; MIP models; multiperiod scenario
1. Introduction
Cutting stock problems appear in a wide range of industrial processes where a variety of large pieces of material, such as paper, glass, steel, wood, or fabric, need to be cut in order to produce smaller pieces of ordered sizes and quantities. Aiming to reduce operating costs, several aspects of the cutting process may be taken into account; and returning leftovers to stock so they can be used in future orders is one of them. See, for example, Koch et al. (2009) and Chen et al. (2019), where the usage of leftovers in the wood-processing and plastic-film industries are considered.
∗Author to whom all correspondence should be addressed (e-mail: [email protected]).
1
In this work, we are concerned with the two-dimensional non-guillotine cutting problem of cutting an heterogeneous set of small rectangular pieces (items) from a set of large rectangular pieces (objects). The problem is said to be two-dimensional because it involves the widths and heights of items and objects;
while it is said to be non-guillotine because cuts are not restricted to be guillotine cuts. A multiperiod scenario is considered in which, at each instantp(0≤p≤P−1), there are ordered items and available purchasable objects; and items ordered at instantpmust be produced within the period[p, p+1]. An item ordered at instantpmay be produced from a purchased object available at that instant or from a leftover of a previously used object. Thus, we consider that, at each instant, an heterogeneous set of objects is available.
In Andrade et al. (2014), the single-period scenario of the problem described in the paragraph above was tackled. In the single-period scenario, the goal is to minimize the cost of the objects required to produce all ordered items; and, within the set of solutions with minimum cost, to maximize the value of objects’usableleftovers. (The formal definition of usable leftover adopted in this work will be given in the next section.) The consideration of usable leftovers assumes their utilization to produce forthcoming orders of items; thus, extending the problem considered in Andrade et al. (2014) to the multiperiod scenario appears as a natural option. In the multiperiod scenario, given a time horizon represented by P instants0, . . . , P −1, the goal is to minimize the overall cost of the objects required to produce all items ordered at instants0, . . . , P −1; and, within the set of solutions with minimum cost, to maximize the value of usable leftovers available at instantP (i.e. at the end of the considered time horizon). It is very clear that it is expected this formulation of the problem to produce better quality solutions than the myopic alternative of solving a single-period problem at each instant. Note that, following Andrade et al. (2014), in the present work there is a clear hierarchy between the two considered objectives—cost must be minimized in first place and, among solutions of minimum cost, a solution with maximum value of usable leftovers is seek. This goals’ hierarchy fits the tackled problem in the bilevel optimization framework Dempe (2002), in opposition to the multiobjective approach Miettinen (1998).
Several works were written regarding one-dimensional cutting problems with leftovers. See the pi- onners’ works Roodman (1986) and Scheithauer (1991), the recent survey Cherri et al. (2014) and the references therein, and the works Poldi and Arenales (2010), Cherri et al. (2013), and Tomat and Gradi˘sar (2017). In Poldi and Arenales (2010), the authors introduce a mixed integer linear programming model, a column generation approach to solve its linear relaxation, and a heuristic rolling horizon approach for rounding off fractional solutions. In Tomat and Gradi˘sar (2017), a multiperiod problem that combines the minimization of the trim-loss and the amount of usable leftovers in stock is considered; and a heuris- tic method is proposed an tested. In Cherri et al. (2013), in order to avoid leftovers remaining in stock for a long period of time, it is considered that the leftovers have a priority-in-use compared to standard objects in stock. A heuristic approach is also proposed and tested.
On the other hand, only a few works address two- and three-dimensional cutting problems with left- overs. In Andrade et al. (2014), a mixed integer linear programming model for the two-dimensional non- guillotine cutting stock problem with usable leftovers was introduced. In the considered single-period problem, the goal is to minimize the cost of the objects required to produce a given set of ordered items and, within the set of minimum cost, maximize the value of the usable leftovers. In Andrade et al. (2016), the non-exact two-stage guillotine cutting stock version of the same problem was analyzed. In Viegas et al. (2016), a heuristic cutting decision process for daily tailored orders of a real-life steel retailer is proposed. The considered problem is a three-dimensional cutting and packing problem in which usable
leftovers of preceding periods may be used to produce items of the current period. A three-staged two- dimensional cutting stock problem with usable leftover is studied in Chen et al. (2015), where an heuristic beam search approach is developed. Exact and non-exact two- and three-stage two-dimensional cutting stock problems are also considered in Silva et al. (2010). Mixed integer linear programming models, that can be seen as extensions of the model proposed in Dyckhoff (1981) for the one-dimensional cutting stock, are proposed. Introduced models are based on the enumeration of all possible ways of producing an item from an object. Since the production of an item from an object produces the item and also two residual objectsthat can be used to generate other items, this work also considers leftovers. Upper bounds on the number of variables and constraints of the proposed models are given. In Silva et al. (2014), the problem introduced in Silva et al. (2010) is extended to the multiperiod framework and integrated with the lot-sizing problem. In this context, the goal is to minimize a total cost that includes raw material, waste, and storage costs. Mixed integer linear programming models and two heuristic approaches based on the industrial practice are proposed. A multiperiod three-dimensional packing problem is addressed in Alonso et al. (2019); in which the problem of putting products on pallets and then loading the pallets into trucks is considered. Mixed integer linear programming models, that include maximum weight con- straints as well as stability constraints, are presented an tested on real instances related to the everyday distribution activity of a company.
The rest of this paper is organized as follows. Section 2 describes and motivates the multiperiod two- dimensional non-guillotine cutting stock problem with leftovers. Section 3 introduces its mixed integer linear programming formulation. In particular, Section 3.6 introduces a model that minimizes the overall cost of the used objects; while Section 3.7 introduces the model that, within the set of solutions with minimum cost, maximizes the value of the leftovers available at the end of the considered time horizon.
Section 4 presents illustrative numerical experiments. Conclusions and lines for future research are given in the last section.
2. The multiperiod two-dimensional non-guillotine cutting stock problem with leftovers
Following Andrade et al. (2014), in this work we consider that object’s leftovers are obtained by per- forming a couple of guillotine pre-cuts on the object that separate the leftovers from the “cutting area”
of the object (region from where the items will be cut). As depicted in Figure 1, there are two possible ways of performing those two cuts: (i) the vertical cut before the horizontal cut or (ii) the horizontal cut before the vertical cut. Given a catalogue of items, we say a leftover is usable if it can fit any item from the catalogue. In this case, the leftover’s value is given by its area times the cost per unit of area of the object. Otherwise, the leftover is disposable and has no value at all. This is why the added values of the two leftovers in Figure 1a may differ from the added values of the two leftovers in Figure 1b. It is worth noting that this definition of leftovers implies that any part of the cutting area of the object that is not used to produce an item is considered waste. (See Andrade et al. (2016) and Andrade et al. (2014) for other definitions of leftovers in two-dimensional problems.)
Assume that there are given (i) a set ofmavailable objectsOjwith widthWj, heightHj, and costcj
per unit of area (j= 1, . . . , m), (ii) a set ofnordered itemsIiwith widthwiand heighthi(i= 1, . . . , n), and (iii) a catalogue composed by ditemsI¯i with widthw¯i and heighth¯i (i = 1, . . . , d). Items from the catalogue are used only to determine whether a leftover is usable or not. Consider the problem of
Potentially valuable top leftover
Potentiallyvaluable righthand-sideleftover Cutting area
Potentially valuable top leftover
Potentiallyvaluable right-handside leftover Cutting area
(a) (b)
Fig. 1. Graphs (a) and (b) represent the two possible ways of generating leftovers performing a vertical and a horizontal guillotine pre-cut. Leftovers are usable if they can fit any item from a given catalogue of items.
cutting all the ordered items from a set of objects of minimum cost. Moreover, assume that when items are cut from an object, two leftovers can be generated (as described above) and that, among all solutions of minimum cost, we want a solution that maximizes the value of the leftovers. This was one of the problems modelled as a mixed integer linear programming problem in Andrade et al. (2014). Clearly, the idea of considering a leftover usable if it can fit an item from the catalogue assumes (a) that new orders will arrive; (b) that new ordered items might be items from the catalogue; and (c) that using the leftovers to cut some of the new ordered items might reduce the cost of purchasing new objects. This suggests the existence of an underlying multiperiod framework.
Consider now P instants of time and assume that, at each instant p (p = 0, . . . , P −1), there are given (i) a set of mp available objectsOpj with width Wpj, height Hpj, and cost cpj per unit of area (j = 1, . . . , mp) and (ii) a set ofnp ordered itemsIpiwith widthwpi and heighthpi (i = 1, . . . , np).
A catalogue composed byditemsI¯i with widthw¯i and height¯hi (i= 1, . . . , d) is also given. At each instantp(p= 0, . . . , P−1), we must decide the way of cuttingallthenpordered items. This means that the cut of ordered items can not be anticipated or delayed. Items may be cut from available objects or from usable leftovers from previous periods. The objective is to minimize the overall cost of the objects required to execute the orders of all instants. Among all solutions of minimum cost, we want a solution that maximizes the value of the leftovers at instantP (the end of the considered time horizon). A leftover is considered valuable if it can fit an item from the catalogue; otherwise it is disposable and it has no value. We assume this problem is part of a larger scenario within which an agent takes the decision of which orders must be placed at each instant considering the demand and existing constraints related to profit, penalties, stock, labor hours, cash flow, etcetera. This means that the presented problem focuses on the determination of the (non-guillotine) cut patterns that minimize the usage of raw material taking advantage of usable leftovers.
Figure 2 illustrates a small instance of the considered problem. Since, all ordered items must be cut, if the objects available at some instant are not enough to produce all items ordered at that instant, the instance may be infeasible. (The possibility of using leftovers from previous periods exists.) To complete the instance, it must be said that the cost per unit of area of all the objects is one and that the catalogue
is composed by the four ordered items. Note that the existence of the catalogue gives some flexibility to the definition of usable leftovers. If, for example, one desires leftovers to be usable whenever they have a minimum width wˆ and a minimum height ˆh then the catalogue may be given by a single item with widthwˆand heightˆh. In this way, leftovers that can fit this item are considered usable and the others are not.
AvailableobjectsOrdereditems
Instantp= 0
O01
6×4
O02
5×5
3×3
Instantp= 1 O11
4×7
O12
4×8
4×5
3×1
Instantp= 2
O21
5×5
2×5
Fig. 2. Illustration a small instance of the considered problem withP= 3periods. The number of available objects at each instant is given bym0=m1= 2andm2= 1and the number of ordered items at each instant is given byn0=n2= 1and n1= 2. The cost per unit of area of all the objects is one (i.e.c01=c02=c11=c12=c21= 1) and the catalogue withd= 4
items is composed by the four ordered items (i.e.w¯1= ¯h1= 3,w¯2= 4,¯h2= 5,w¯3= 3,¯h3= 1,w¯4 = 2, and¯h4 = 5).
Figure 3 illustrates three different feasible solutions to the instance in Figure 2. Figure 3a represents a solution that can be found by a myopic approach that proceeds as follows. At each instant, available objects are the objects that can be bought and also the usable leftovers from previous periods (with no cost). The solution to each instant can be found by solving the model introduced in Andrade et al. (2014) that consists in minimizing the cost of the objects required to cut the ordered items and, among solutions with minimum cost, chooses one with maximum value of the usable leftovers. The solution in Figure 3a uses objectsO01,O11, andO21whose total cost is24 + 28 + 25 = 77and, at instantp = 3, has three remaining usable leftovers whose total value is given by 12 + 8 + 15 = 35. Solutions in Figures 3b and 3c use objectsO02 andO11 only, whose total cost is25 + 28 = 53, implying that the solution in Figure 3a is not optimal. Note that the smaller total cost of both solutions was obtained by buying a more expensive object at instant p = 0. Moreover, the solution in Figure 3b has usable leftovers at instant p= 3whose total value is6 + 4 = 10; while Figure 3c has usable leftovers at instantp= 3whose total value is3 + 8 = 11, being in fact the optimal solution we are looking for.
3. Mixed integer linear programming formulation
In this section, a mixed integer linear programming formulation for the multiperiod two-dimensional non-guillotine cutting problem with usable leftovers, described in the previous section, is given. The formulation is an extension of the single-period model considered in Andrade et al. (2014) and the novelty relies on the existence, on a given period, of objects that are usable leftovers of previous periods.
The dimensions of that objects are not constants but depend on the cutting patterns of the previous periods.
3.1. Instance data
LetP be the number of instants to be considered. Assume that, for eachp = 0, . . . , P −1, there are given (a)mp ≥ 0and a set ofmp available objectsOpj with widthWpj, height Hpj, and costcpj per unit of area (j = 1, . . . , mp) and (b)np ≥0and a set ofnpordered itemsIpiwith widthwpiand height hpi (i = 1, . . . , np). A catalogue composed byditemsI¯i with widthw¯i and height¯hi (i = 1, . . . , d) is also given. As explained in the previous section, each object available at instant p generates two leftovers (that may be usable or not) at instantp+ 1; those two leftovers generate two leftoverseachat instantp+ 2; and so on. This means that after ∆p instants there will be2∆p leftovers associated with each object of instantp. This fact would make intractable (to be solved to global optimality) instances with even moderate values ofP. Therefore, from the theoretical point of view, it makes sense to add to the problem an integer parameterξ ∈[0, P]that says for how many periods leftovers of an object may be available. Ifξ = 0then the problem considers no leftovers at all. Ifξ = P then all leftovers will be available up to instantP. Parameterξis considered to be the same for all objects of all instants with the only purpose of simplifying the presentation. In practice, each object of each instant might have its own
“duration” parameterξthat would represent the perishability of the raw material it is made of.
3.2. Additional computable constants
By definition, each object generates two leftovers; and leftovers generated in a period[p, p+ 1]remain available up to period[p+ξ, p+ξ+ 1];ξ = 0meaning that leftovers are not being considered at all.
This means that the numberm¯p ≥ mp of available objects at a given periodp, composed by themp purchasable objects plus the objects that are leftovers of previous periods, is given by
¯ mp =
mp, p= 0,
mp+ leftovers(p, ξ), p= 1, . . . , P −1, leftovers(p, ξ), p=P,
(1)
where
leftovers(p, ξ) =
min{p,ξ}
X
`=1
2`mp−`.
Note that, since, by definition, there are no purchasable objects at instantP,m¯P represents thenumber of leftoversavailable at instantP only.
By definition, the cost (or value) per unit of area of a leftover is the cost per unit of area of the object that originated the leftover. We aim to define¯cpj as the cost per unit of area of each objectOpj (p = 0, . . . , P, j = 1, . . . ,m¯p). If we name Op+1,j1 and Op+1,j2, withj1 = mp+1 + 2j −1 and j2=mp+1+ 2j, the leftovers generated by objectOpj(p= 0, . . . , P−1,j= 1, . . . ,m¯p) then we have that
¯
cp+1,j1 = ¯cp+1,j2 = ¯cpj =cpjforp= 0, . . . , P −1, j = 1, . . . , mp.
The relevant costs that will be used later in this section are the costs¯cP j(j = 1, . . . ,m¯P) that correspond to the value (per unit of area) of the leftovers available at instantP; i.e. at the end of the considered time horizon.
3.3. Assignment of items to objects
Letvpij ∈ {0,1}(p = 0, . . . , P −1,i = 1, . . . , np,j = 1, . . . ,m¯p) be such thatvpij = 1if itemIpi must be produced from objectOpj andvpij = 0, otherwise. The fact that each item must be produced from exactly one object can be modeled with the constraints
¯ mp
X
j=1
vpij = 1, p= 0, . . . , P−1, i= 1, . . . , np. (2)
Let upj ∈ {0,1} (p = 0, . . . , P −1,j = 1, . . . ,m¯p) be such that upj = 1 if at least one item is produced from the objectOpjandupj = 0, otherwise. This can be modeled with the constraints
upj ≥vpij, p= 0, . . . , P −1, j= 1, . . . ,m¯p, i= 1, . . . , np (3) and
upj ≤
np
X
i=1
vpij, p= 0, . . . , P −1, j= 1, . . . ,m¯p. (4)
3.4. Objects’ dimensions
LetW¯pj andH¯pj (p = 0, . . . , P,j = 1, . . . ,m¯p) be the width and height of objectOpj, respectively.
Clearly, for everypandj ≤mp, we have thatW¯pj =Wpj andH¯pj =Hpj. We now turn our attention to the dimensionsW¯pjandH¯pjforj∈ {mp+1, . . . ,m¯p}, i.e. dimensions of objectsOpjwithp >0that are leftovers from previous periods. Assuming that the dimensions of all objects of a given instantpare already known (and this is true forp = 0), we show how to determine the dimensions of the objects of instantp+ 1.
Letp∈ {0, . . . , P −1}be an instant and lettpjandrpj (j= 1, . . . ,m¯p), satisfying
0≤tpj≤H¯pjand0≤rpj ≤W¯pj, j= 1, . . . ,m¯p, (5)
be such thatH¯pj−tpjandW¯pj−rpjare the height and the width, respectively, of the “cutting area” of objectOpj(see Figure 1); and letηpj ∈ {0,1}be such thatηpj = 1if the vertical pre-cut is made in first place (see Figure 1a) andηpj = 0otherwise (see Figure 1b).
With the help of these three variables (tpj,rpj, andηpj), we are able to determine the dimensions of the two leftovers generated bythe usage or notof objectOpj, that correspond to two available objects at instantp+ 1. As it can be seen in Figure 1, if the objectis usedandηpj = 1then the leftover at the top of the object has widthW¯pj−rpj and heighttpj; while the leftover on the right hand side of the object has widthrpj and heightH¯pj. On the other hand, if the objectis usedandηpj = 0then the leftover at the top of the object has widthW¯pjand heighttpj; while the leftover on the right hand side of the object has widthrpjand heightH¯pj−tpj. When the objectis not used, we must consider in separate the case in which the object is a purchasable object (1 ≤j ≤mp) and the case in which the object is a leftover from a previous period (mp+ 1≤j≤m¯p).
In the case of an unused purchasable object Opj, since an object that is not purchased generates no leftovers, we must define the dimensions of its leftovers as null. Thus, we can model the dimensions of the leftovers of a purchasable object as
0 ≤ H¯p+1,j1 ≤ Huˆ pj,
tpj−(1−upj) ˆH ≤ H¯p+1,j1 ≤ tpj+ (1−upj) ˆH, 0 ≤ W¯p+1,j1 ≤ W uˆ pj,
W¯pj−rpj−(1−ηpj) ˆW −(1−upj) ˆW ≤ W¯p+1,j1 ≤ W¯pj−rpj+ (1−ηpj) ˆW + (1−upj) ˆW , W¯pj−ηpjWˆ −(1−upj) ˆW ≤ W¯p+1,j1 ≤ W¯pj+ηpjWˆ + (1−upj) ˆW ,
0 ≤ W¯p+1,j2 ≤ W uˆ pj,
rpj−(1−upj) ˆW ≤ W¯p+1,j2 ≤ rpj+ (1−upj) ˆW , 0 ≤ H¯p+1,j2 ≤ Huˆ pj,
H¯pj−(1−ηpj) ˆH−(1−upj) ˆH ≤ H¯p+1,j2 ≤ H¯pj+ (1−ηpj) ˆH+ (1−upj) ˆH, H¯pj−tpj−ηpjHˆ −(1−upj) ˆH ≤ H¯p+1,j2 ≤ H¯pj−tpj+ηpjHˆ + (1−upj) ˆH, (6)
forj = 1, . . . , mp, wherej1 =mp+1+ 2j−1,j2 =mp+1+ 2j, and the constantsWˆ andHˆ are given byWˆ = max{Wpj |p = 0, . . . , P −1, j = 1, . . . , mp}andHˆ = max{Hpj |p = 0, . . . , P −1, j = 1, . . . , mp}.
In the case of an unused object Opj that is a leftover from a previous period, we must have an ob- jectOp+1,j1 identical toOpj and an objectOp+1,j2 with null dimensions; or the analogous situation in which objectsOp+1,j1 andOp+1,j2 change their places. If we arbitrary consider the first case, we can
model the dimensions of the leftovers of an object that is a leftover from a previous period as H¯pj−Huˆ pj ≤ H¯p+1,j1 ≤ H¯pj+ ˆHupj,
tpj−(1−upj) ˆH ≤ H¯p+1,j1 ≤ tpj+ (1−upj) ˆH, W¯pj−W uˆ pj ≤ W¯p+1,j1 ≤ W¯pj+ ˆW upj,
W¯pj−rpj−(1−ηpj) ˆW −(1−upj) ˆW ≤ W¯p+1,j1 ≤ W¯pj−rpj+ (1−ηpj) ˆW + (1−upj) ˆW , W¯pj−ηpjWˆ −(1−upj) ˆW ≤ W¯p+1,j1 ≤ W¯pj+ηpjWˆ + (1−upj) ˆW ,
0 ≤ W¯p+1,j2 ≤ W uˆ pj,
rpj−(1−upj) ˆW ≤ W¯p+1,j2 ≤ rpj+ (1−upj) ˆW , 0 ≤ H¯p+1,j2 ≤ Huˆ pj,
H¯pj−(1−ηpj) ˆH−(1−upj) ˆH ≤ H¯p+1,j2 ≤ H¯pj+ (1−ηpj) ˆH+ (1−upj) ˆH, H¯pj−tpj−ηpjHˆ −(1−upj) ˆH ≤ H¯p+1,j2 ≤ H¯pj−tpj+ηpjHˆ + (1−upj) ˆH, (7) forj =mp+ 1, . . . ,m¯p. Note that (6) and (7) differ only in the constraints that apply toW¯p+1,j1 and H¯p+1,j1 whenupj = 0. While (6) says that in this case we must haveW¯p+1,j1 = ¯Hp+1,j1 = 0; (7) says that it must holdW¯p+1,j1 = ¯WpjandH¯p+1,j1 = ¯Hpj.
A technicality is missing and, therefore, there is some abuse of notation in the description of con- straints (6) and (7). In both constraints, it is assumed that every object generates two leftovers (as it is in fact whenξ =P). Thus, it is written that constraint (6) applies to allj = 1, . . . , mp, constraints (7) applies to all j = mp + 1, . . . ,m¯p, and we define j1 = mp+1 + 2j −1 andj2 = mp+1 + 2j. In practice, every objectOpj has an associated “shelf life”. A purchasable object has shelf lifeξ; while an object that is a leftovers has a shelf life that is one less than the shelf life of the object that generated the leftover. Then, only objects with a strictly positive shelf life generate leftovers; and the leftovers must be numbered accordingly. For example, if, at a periodp,Op,ja,Op,jb, . . . (withja ≤jb ≤ . . .) are the objects that generate leftovers (i.e. the objects with a strictly positive shelf life) then the two leftovers of Op,jashould be numberedmp+1+ 1andmp+1+ 2; while the two leftovers ofOp,jbshould be numbered mp+1 + 3and mp+1+ 4. In any case, note that the number of leftovers at every period p, given by
¯
mp−mp, wherem¯pis defined in (1), is fixed and it depends only on the instance data and the additional computable constants described in Sections 3.1 and 3.2.
3.5. Avoiding overlapping and fitting items within objects’ “cutting area”
We now consider the positioning constraints that avoid overlapping of items produced from the same object and the constraints that fit the items within the cutting area of the objects. For this, let(xpi, ypi) be the Cartesian coordinates of the center of itemIpi (p = 0, . . . , P −1,i = 1, . . . , np). The fitting constraints, given by
0≤xpi−12wpiandxpi+12wpi≤W¯pj−rpj+ (1−vpij) ˆW ,
0≤ypi−12hpiandypi+ 12hpi≤H¯pj−tpj+ (1−vpij) ˆH, (8)
forp= 0, . . . , P −1,i= 1, . . . , np,j = 1, . . . ,m¯p, say that itemIpimust be placed within the cutting area of the objectOpj to which it was assigned. Note that we have assumed, without loss of generality, that the bottom-left corner of all objects corresponds to the origin of the Cartesian plane.
LetIpiandIpi0 withi0 > ibe two items that are being assigned to the same objectOpj, i.e.vpij = vpi0j = 1. The non-overlapping constraints must say that|xpi−xpi0| ≥ 12(wpi+wpi0)or|ypi−ypi0| ≥
1
2(hpi+hpi0). Using that|a| ≥bis the same thata≥bor−a≥b, these disjunction can be written using their big-M formulation as
xpi−xpi0 ≥ 12(wpi+wpi0) − Wˆ [(1−vpij) + (1−vpi0j) +πpii0+τpii0],
−xpi+xpi0 ≥ 12(wpi+wpi0) − Wˆ [(1−vpij) + (1−vpi0j) +πpii0+ (1−τpii0)], ypi−ypi0 ≥ 12(hpi+hpi0) − Hˆ [(1−vpij) + (1−vpi0j) + (1−πpii0) +τpii0],
−ypi+ypi0 ≥ 12(hpi+hpi0) − Hˆ [(1−vpij) + (1−vpi0j) + (1−πpii0) + (1−τpii0)], (9)
forp= 0, . . . , P−1,j = 1, . . . ,m¯p,i= 1, . . . , np,i0 =i+ 1, . . . , np, whereπpii0andτpii0 ∈ {0,1}for p= 0, . . . , P−1,i= 1, . . . , np, andi0 =i+ 1, . . . , npare auxiliary variables. When two items ordered at the same instant are identical (i.e. they have identical dimensions), additional constraints (introduced in (Andrade and Birgin, 2013,§3)) may be added in order to avoid symmetric solutions in which both items interchange their places. See also (Andrade et al., 2014, p.1651).
3.6. Minimizing the cost of the used objects
Up to this point, we have all the elements to build up the mixed integer linear programming formulation of the problem of minimizing the overall cost of the objects required to satisfy the orders of all instants making use of leftovers. Variables of the problem arevpij ∈ {0,1}(p= 0, . . . , P −1,j = 1, . . . ,m¯p, i= 1, . . . , np),upj ∈ {0,1}, (p= 0, . . . , P −1,j = 1, . . . , mp),ηpj ∈ {0,1},W¯pj,H¯pj,tpj,rpj ∈R (p = 0, . . . , P −1, j = 1, . . . ,m¯p), and πpii0, τpii0 ∈ {0,1} (p = 0, . . . , P −1, i = 1, . . . , np, i0=i+ 1, . . . , np). Since the cost of the used objects is given by
P−1
X
p=0 mp
X
j=1
cpjWpjHpjupj, (10)
the problem is given by minimizing (10) subject to the constraints (2), (3), and (4) that assign items to objects, the constraints (5), (6), and (7), that determine the dimensions of the leftovers, and the constraints (8) and (9) that avoid overlapping between the items and fit the items within the cutting area of the objects, respectively.
3.7. Maximizing the value of usable leftovers at the end of the time horizon
If we consider the instance depicted in Figure 2, solutions illustrated in Figures 3b and 3c are both optimal solutions to the model introduced in Section 3.6. This is because, although leftovers are being used to reduce the cost of the required objects; the value of the usable leftovers available at instantPare notbeing consider in the model to determine that, in fact, the solution in Figure 3c is preferred. Thus, we now need to model that, by definition,usableleftovers are the ones that can fit at least an item from the catalogue, while the other ones are disposable; and that, also by definition, the value of an usable leftover is given by its area times the cost per unit of area of the object that generated the leftover. Then, the value of the usable leftovers available at instantP must be incorporated into the model as a tie break to differentiate solutions with minimum cost of the used objects.
All objects available at instantP are, by definition, leftovers of previous periods, since there are no purchasable objects at this (last) instant. Moreover, they are exactlym¯P objects and they have widthW¯P j
and heightH¯P j forj = 1, . . . ,m¯P. Each objectOP j that can fit an item from the catalogue has value
¯
cP jW¯P jH¯P j; while the others have no value and are disposable. Assume we are able to introduce vari- ablesγj (j = 1, . . . ,m¯P) and additional constraints that impose thatγj = ¯WP jH¯P j if there exists an itemIi (i= 1, . . . , d) from the catalogue such thatw¯i≤W¯P j and¯hi≤H¯P j; andγj = 0otherwise. In this case, the value of the usable leftovers at instantP is given by
¯ mP
X
j=1
¯ cP jγj.
Since leftovers come from objects, the value of all leftovers is strictly smaller than the cost of the objects they come from. Thus,
¯ mP
X
j=1
¯ cP jγj ≤
¯ mP
X
j=1
¯
cP jW¯P jH¯P j <
P−1
X
p=0 mp
X
j=1
cpjWpjHpjupj≤
P−1
X
p=0 mp
X
j=1
cpjWpjHpj.
If we assume that cpj, Wpj, and Hpj (p = 0, . . . , P −1, j = 1, . . . , mp) are integer numbers then minimizing
P−1
X
p=0 mp
X
j=1
cpjWpjHpjupj− 1 PP−1
p=0
Pmp
j=1cpjWpjHpj
!m¯
P
X
j=1
¯ cP jγj
or, equivalently,
P−1
X
p=0 mp
X
j=1
cpjWpjHpj
P−1
X
p=0 mp
X
j=1
cpjWpjHpjupj
−
¯ mP
X
j=1
¯
cP jγj (11)
has the effect of minimizing the overall cost of the used objects and, within the set of solutions with minimum cost, maximizing the value of the usable leftovers available at instantsP.
3.7.1. Modelling the area of the usable leftovers
It remains to describe the constraints that produce the desired effect on the variablesγj(j= 1, . . . ,m¯P), i.e.
γj =
W¯P jH¯P j, if there existsi∈ {1, . . . , d}such thatw¯i ≤W¯P j and¯hi ≤H¯P j,
0, otherwise.
If, in addition to the integrality of cpj,Wpj, andHpj (p = 0, . . . , P −1,j = 1, . . . , mp), we also assume that all ordered items have integer dimensions, i.e. that wpi andhpi (p = 0, . . . , P −1, j = 1, . . . , mp) are all integers, then we have that there are optimal solutions for whichW¯P j andH¯P j (j = 1, . . . ,m¯P) are all integer as well (see, for example, Birgin et al. (2010, 2012)). Therefore, we can expressW¯P jas
W¯P j =
L
X
`=1
2`−1θj`, (12)
whereL=blog2( ˆW)c+ 1andθj`∈ {0,1}(j= 1, . . . ,m¯P,`= 1, . . . , L), i.e.θjLθj,L−1. . . θj1being the binary representation ofW¯P j andLbeing an upper bound on the number of required bits. With this, we have that, forj= 1, . . . ,m¯P,
W¯P jH¯P j =
L
X
`=1
2`−1H¯P jθj`;
thus representing the product of two integers by the sum of products of an integer and a binary vari- able (see, for example, Harjunkoski et al. (1997); Yanasse and Morabito (2006)). The value of each of these products coincides with the value of the integer if the binary variable is one; and zero otherwise.
Introducing (continuous) variablesωj`(j = 1, . . . ,m¯P,`= 1, . . . , L), this products can be modelled as
0≤ωj`≤H¯P j andH¯P j−(1−θj`) ˆH≤ωj`≤θj`Hˆ forj= 1, . . . ,m¯P, `= 1, . . . , L. (13) Up to now, we have that, with the variablesθj`∈ {0,1}andωj`(j= 1, . . . ,m¯P,`= 1, . . . , L) and the constraints (13), each productW¯P jH¯P j(j= 1, . . . ,m¯P) is given by
L
X
`=1
2`−1ωj`.
Now consider variablesζji ∈ {0,1}(j = 1, . . .m¯P,i= 1, . . . , d). The idea is thatζji = 0if itemIi from the catalogue does not fit within objectOP j, i.e. ifw¯i >W¯P j or¯hi > H¯P j. We model this with the constraints
¯
wi≤W¯P j+ ˆW(1−ζji)and¯hi ≤H¯P j+ ˆH(1−ζji)forj= 1, . . . ,m¯P, i= 1, . . . , d. (14)
Now, constraints 0≤γj ≤
L
X
`=1
2`−1ωj`andγj ≤
d
X
i=1
ζji
!
WˆHˆ forj= 1, . . . ,m¯P (15)
say that the value of γj may vary between zero and the area of object OP j; and that γj = 0 if no item from the catalogue fits within objectOP j. Since eachγj appears with a negative coefficient in the objective function being minimized, in any solution we will haveγjequal to its maximum possible value as desired.
3.7.2. The full model
Summing up, we now have all the ingredients to build up the complete mixed integer linear programming formulation of the problem of minimizing the overall cost of the objects required to satisfy the demand of instants from 0 to P −1 making use of leftovers; and, among all solutions with minimum cost, maximizing the value of the usable leftovers at instantP.
Variables of the problem arevpij ∈ {0,1}(p= 0, . . . , P −1,j = 1, . . . ,m¯p,i= 1, . . . , np),upj ∈ {0,1}(p = 0, . . . , P,j = 1, . . . , mp),ηpj ∈ {0,1},tpj,rpj ∈ R(p = 0, . . . , P −1,j = 1, . . . ,m¯p), W¯pj,H¯pj (p = 0, . . . , P,j = 1, . . . ,m¯p),πpii0,τpii0 ∈ {0,1}(p = 0, . . . , P −1,i = 1, . . . , np,i0 = i+ 1, . . . , np),γj,θj` ∈ {0,1},ωj` (j = 1, . . . ,m¯P,`= 1, . . . , L), andζji ∈ {0,1}(j = 1, . . . ,m¯P, i= 1, . . . , d).
The problem is given by minimize (11) subject to the constraints (2), (3), and (4), that assign items to objects, the constraints (5), (6), and (7), that determine the dimensions of the leftovers, the constraints (8) and (9) that avoid overlapping between the items and fit the items within the cutting area of the objects, respectively, and the constraints (13), (14), and (15) that model the value of the usable leftovers at instantP. (Note that, due to (12), or (12) forj = 1, . . . ,m¯P is added to the model or variablesW¯P j (j = 1, . . . ,m¯P) are eliminated as variables and all their occurrences are replaced by the right hand side of (12). In the numerical experiments included in the next section, we arbitrarily opted by including (12) forj= 1, . . . ,m¯P as constraints of the model.)
4. Illustrative numerical examples
In this section, we present numerical experiments with the proposed model. The goal of the numerical experiments is to analyze the influence of considering leftovers in the overall cost of the purchased objects; so each considered instance will be solved varyingξ ∈ {0,1, . . . , P}. Recall thatξ = 0means that leftovers are not considered at all; whileξ = P means that leftovers generated at any period are available up to the end of the considered time horizon. Twenty five small-sized instances with up to four periods will be solved with an exact commercial solver. Tables 1 and 2 describe the instances. In the tables, for each instance,P is the number of periods. For each instantp = 0,1, . . . , P −1, mp is the number of purchasable objects andnp is the number of ordered items. Notationa(b×c)[s]means that there areaobjects or items with widthband heightc; and, in the case of objects, that the cost per unit of area iss. Whenais omitted, it means that there is a single copy of the described object or item; and, whensis omitted, it means that the cost per unit of area is1. In the last column, dis the number of
items in the catalogue. Items in the catalogue are the ones whose dimensions are underlined in the table.
Table 3 displays the number of binary and continuous variables and the number of constraints of each one of the twenty five considered instances varyingξ ∈ {0,1, . . . , P}. From the table, it is easy to see how these figures grow as a function ofξ.
The model was implemented in C/C++ using the ILOG Concert Technology and compiled with g++
from gcc version 5.4.0 (GNU compiler collection) with the -O3 option enable. Numerical experiments were conducted using a machine with Intel Xeon Processor X5650, 8GB of RAM memory, and Ubuntu 16.04 operating system. Instances were solved using IBM ILOG CPLEX 12.8.0. By default, a solution is reported as optimal by the solver when
absolute gap=best feasible solution−best lower bound≤εabs
or
relative gap= |best feasible solution−best lower bound| 10−10+|best feasible solution| ≤εrel,
with εabs = 10−6 and εrel = 10−4, where “best feasible solution” means the smallest value of the objective function related to a feasible solution generated by the method. The objective function (11) has the particular property of assuming relatively large integer values at feasible points. Hence, a stopping criterion based on a relative error less than or equal toεrel = 10−4 may have the undesired effect of stopping the method prematurely. On the other hand, due to the integrality of the objective function values, an absolute error strictly smaller than 1 is enough to prove the optimality of the incumbent solution. Therefore, in the numerical experiments, we considered εabs = 1−10−6 and εrel = 0. In addition,NodeFileIndandWorkMem parameters were set to3and6,000, respectively; so the Branch
& Bound tree is partially transferred to disk if memory is exhausted. All other parameters of the solver were used with their default values.
Tables 4, 5, and 6 describe the solutions found to the twenty five considered instances for varying values ofξ ∈ {0,1},ξ ∈ {2,3}, and ξ = 4, respectively. In the tables, “Objective function optimal value” is the value of the objective function (11) at the solution reported as optimal by the solver. “Objects costs” and “Leftovers value” correspond to the cost of the purchased objects and the value of the leftovers at instantP, respectively; and they are extracted from the optimal value according to (11). A CPU time limit of two hours was imposed to the solver. When this time limit is reached, as it is the case in some instances with ξ = 4, the “best lower bound”, “the best feasible solution”, and the “gap in %” are reported instead of the unknown optimal value. Remaining columns “MIP iterations”, “B&B Nodes”, and “CPU time” (in seconds) are self-explanatory and state the effort required by the solver to obtain the reported solution. Figure 4 shows the influence of considering leftovers in the reduction of the cost of the objects that need to be purchased to satisfy the overall demand of items along the considered time horizon. As expected, the use of leftovers significantly reduces the cost of the purchased objects.
It should be noted that for ξ = 3andξ = 4 the costs of the purchased objects coincide in all the25 instances. The difference in these two scenarios relies on the value of the remaining leftovers. When ξ = 3there are remaining leftovers in five instances only (namely, instances1,2,3,5, and22). When ξ = 4there are remaining leftovers in all instances but instance 4. It can be highlighted the case of instance14that has no remaining leftovers when ξ = 3and it has remaining leftovers with value582
whenξ= 4.
As it can be seen in Table 4, instances4, 5, and9are infeasible whenξ = 0. This is because these instances have instants with ordered items and no available objects; andξ= 0impair the use of leftovers.
The same happens with instance9for the caseξ = 1since it has two consecutive periods with ordered items and no available objects. Tables 4 and 5 show that forξ∈ {0,1,2,3}the solver was able to detect infeasibility or to find an optimal solution for all the 25instances, within the CPU time limit. For the caseξ = 4(see Table 6), the CPU time limit was reached for instances6,9,10,14,15,18,23, and25.
However, it should be noted that the final gap was larger than1%in only one instance. The median CPU times forξ∈ {0,1,2,3,4}are0.01,1.36,4.69,2.92, and204.76, respectively.
Figures 5–7 show the graphical representation (cutting/packing patterns) of the solutions obtained for (the arbitrarily chosen) instances 6, 7, and8, varying ξ ∈ {0,1,2,3,4}. Figure 5a shows that, when ξ = 0, i.e. when leftovers are not considered, four objects that cost1,177are needed to cut the ordered items; and, of course, there are no leftovers at the end of time horizon. Whenξ = 1, leftovers can be used only in the period that follows the period in which they were generated. Figure 5b shows that, in this case, only two objects that cost716are required. One is bought at instantp = 0and the other at instantp= 2. The first one is used to cut the items ordered at instantp= 0and it generates a leftover that is used to cut the items ordered at instantp= 1. The same happens at instantp= 2, where an object is bought that is used to cut the items ordered at that instant and generates two leftovers that are used to cut the items ordered at instantp= 3. Since no object is bough at instantp = 3andξ= 1, there are no leftovers at the end of the time horizon. The caseξ = 2is depicted in Figure 5c. In this case, the leftovers of the object bought at instantp = 0are used to cut the items ordered at instantsp = 1andp = 2. A new object is bought at instantp = 3and two leftovers remain at the end of the time horizon. The two purchased objects cost 595. When ξ = 3 (see Figure 5d), a single object that costs374is bought at instantp= 0. This object and its leftovers are enough to cut all ordered items (at instantsp= 0,1,2,3).
Sinceξ= 3and the only object is bought at instantp= 0, there are no leftovers remaining at the end of the time horizon. Whenξ = 4(see Figure 5e) the same object is bought at instantp= 0, but the cutting pattern is chosen in order to maximize the value of the leftovers remaining at the end of the time horizon.
Figures 6 and 7 correspond to instances7and8, respectively. These two instances are very similar, the only difference being that they have the cost per unit of area of the two objects with dimensions10×12 and12×10that are available at instantp= 0interchanged. The figures show that, in the caseξ= 4, the cutting pattern is such that the value of the remaining leftovers at the end of the horizon is maximized.
And this is achieved concentrating the leftovers in the object with a larger cost per unit of area. Note that, in instance7, the overall area of the remaining leftovers is95, but its value is190(see Table 6); while in instance8the overall leftovers’ area is107with a value of175(see Table 6).
As it can be seen in Tables 1 and 2, instance 1 has three periods; while all the other twenty four considered instances have four periods. It is very clear from the problem formulation that the number binary variables in the proposed model depends on the number of periods P and on the number of available objectsm¯p and ordered itemsnpat each instantp= 0, . . . , P−1. Moreover, since, as defined in (1), the number of available objectsm¯pat a given instantpcorresponds to the number of purchasable objects mp plus the number usable leftovers from previous periods, m¯p depends exponentially on p and on parameterξthat says for how many periods leftovers of previous periods may be available. This dependency is illustrated in Table 3. Moreover, the results in Tables 4, 5, and 6 illustrate that, as expected, the solver’s effort increases as a function ofξ and that several instances withξ = 4can not be solved to
optimality within the CPU time limit of two hours. It should also be stressed that, as already mentioned, the goal of the numerical experiments was to analyze the influence of considering leftovers in the overall cost of the purchased objects. Therefore, the set of instances was chosen in such a way that the solver were able to find optimal solution within an affordable time limit; while it is not a surprise the solver would not be able to find optimal solutions within a reasonable amount of time for much larger instances.
5. Concluding remarks
Two-dimensional non-guillotine cutting stock problems with leftovers in which leftovers can be gen- erated by two guillotine pre-cuts were considered in this work. In particular, the cutting problem with leftovers introduced in Andrade et al. (2014) was embedded into a multiperiod framework. In this way, objects and leftovers at each period can be better chosen in order to minimize the overall cost of the objects that are required to execute a given set of sorted orders. Some alternative variants of the problem were analyzed. A MIP formulation of the considered problem was introduced and illustrative numerical experiments were presented. On the one hand, since, as expected, practical instances could not be solved to optimality with an exact solver, developing heuristic methods for the introduced problem would be a possible direction for future research. On the other hand, considering uncertainty in the problem data would make the problem closer to practice. A more ambitious goal would be to integrate this multi- period cutting problem with leftovers with a lot sizing problem in order to simultaneously determine the demanded items that must be placed as orders and their optimal cutting pattern.
Acknowledgments
This work has been partially supported by FAPESP (grants 2013/03447-6, 2013/05475-7, 2013/07375-0, and 2016/01860-1) and CNPq (grants 146110/2013-7, 309517/2014-1 and 306083/2016-7). The authors would like to thank the reviewers for their helpful comments.
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Instant p= 0
Period[0,1]
O01
O02
3×3
Instant p= 1
Period[1,2]
O11
O12
4×5
3×1 Instant
p= 2
Period[2,3]
O21
2×5
Instant p= 3
Remaining usable leftovers
3×4
3×5 4×2
(a)
Instant p= 0
Period[0,1]
O01 O02 3×3
Instant p= 1
Period[1,2]
O11 O12
4×5 3×1
Instant p= 2
Period[2,3]
O21
2×5
Instant p= 3
Remaining usable leftovers
3×2
4×1
(b)
Instant p= 0
Period[0,1]
O01
O02
3×3
Instant p= 1
Period[1,2]
O11
O12
4×5
3×1
Instant p= 2
Period[2,3]
O21
2×5
Instant p= 3
Remaining usable leftovers
3×1
4×2
(c)
Fig. 3. Illustration of solutions that, at each period, may cut ordered items from usable leftovers from previous periods. (a) Greedy solution obtained by a myopic method that,at each instant, minimizes the cost of the objects required to cut the ordered items of that instant, assuming that usable leftovers from previous periods are free. (b) Solution that minimizes the overall cost of the required objects. (c) Solution with minimum cost of the required objects and, in addition, maximum value of
the usable leftovers at instantp= 3.
